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Λ2-Weighted statistical convergence and Korovkin and Voronovskaya type theorems
Applied Mathematics and Computation 266 (2015) 675–686
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2 -Weighted statistical convergence and Korovkin and Voronovskaya type theorems Naim L. Braha a, Valdete Loku a, H.M. Srivastava b,∗ a Department of Computer Sciences and Applied Mathematics, College Vizioni Për Arsim, Street Ahmet Kaciku Nr. 3, Kati i dyte, Ferizaj 70000, Republic of Kosova b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
a r t i c l e
i n f o
a b s t r a c t In this paper, we propose to introduce a new 2 -weighted statistical convergence. Based upon this definition, we prove some Korovkin type theorems. We also find the rate of the convergence for this kind of weighted statistical convergence and derive some Voronovskaya type theorems.
MSC: Primary 40C05 40G15 41A36 Secondary 46A35 46A45 46B45
© 2015 Elsevier Inc. All rights reserved.
Keywords: Statistical convergence and statistical summability 2 -Weighted statistical convergence and Korovkin type theorems Rate of convergence and Voronovskaya type theorems Positive linear operators Bounded and continuous functions and modulus of continuity Nonincreasing and nondecreasing functions
1. Introduction, definitions and preliminaries We denote by N the set of all natural numbers. We also let K ⊂ N and suppose that
Kn := {k : k n and
k ∈ K}.
Then the natural density of K is defined by
d (K ) := lim
n→∞
|Kn | n
= lim
n→∞
1 |{k : k n and n
k ∈ K}|
if the limit exists. Here, and in other similar situations, by || we denote the number of elements in the enclosed set . A given sequence X = (xn ) is said to be statistically convergent to L if, for each > 0, the set
K := {k : k ∈ N ∗
and
|xk − L| }
Corresponding author. Tel.: +1 250 472 5313; fax: +1 250 721 8962. E-mail addresses: [email protected] (N.L. Braha), [email protected] (V. Loku), [email protected] (H.M. Srivastava).
http://dx.doi.org/10.1016/j.amc.2015.05.108 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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has natural density zero (see [7] and [17]), that is, for each > 0, we have
lim
n→∞
|K | n
= lim
n→∞
1 |{k : k n and n
|xk − L| }| = 0.
In this case, we write
L = stat limn→∞ xn . We note here that every convergent sequence is statistically convergent, but not conversely. be a strictly increasing sequence of positive real numbers which tend to infinity as n → ∞, that is, let Let = (λn )∞ n=0
0 < λ0 < λ1 < · · · < λn < · · · ,
lim
n→∞
λn = ∞ and 2 (λn ) 0,
where the first difference (λk ) and the second difference 2 (λk ) are defined by
λk = λk − λk−1 (λ−1 = λ−2 = 0 ) and
2 (λk ) = ((λk )) = λk − 2λk−1 + λk−2 (λ−1 = λ−2 = 0 ), respectively. Let X = (xk ) be a sequence of complex numbers such that x−1 = x−2 = 0. We then denote by 2n (x ) the following sum (see [3]):
2n (x ) =
n 1 (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ). λn − λn−1 k=0
Definition 1. The sequence (xn ) is said to be 2 -summable to L if
lim 2n (x ) = L.
n→∞
We also say that the sequence (xn ) is statistically summable to L by the weighted method determined by the sequence 2n (x ) if
stat limn→∞ 2n (x ) = L. We denote by 2 (stat) the set of all sequences which are 2 -statistically summable. Definition 2. We say that the sequence (xn ) is weighted 2 -statistically convergent to L if, for every > 0,
lim
n→∞
1
λn − λn−1
|{k λn − λn−1 : |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| }| = 0.
In this case, we write
L = stat2 limn→∞ xn . Definition 3. A sequence X = (xn ) is said to be strongly 2r -summable (0 < r < ∞) to the limit L if
lim
n→∞
n 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L|r = 0. λn − λn−1 k=1
We write it as xk → 2r . In this case, L is called strongly 2r limit of the sequence X = (xn ). Based upon the above definition of the 2 -weighted statistical convergence, we propose to prove here some Korovkin type theorems. We also find the rate of the convergence for this kind of weighted statistical convergence and derive some Voronovskaya type theorems. 2. Relations involving the 2 -statistically summable and convergent sequences and strongly 2r -summability sequences Motivated essentially by several recent works (see, for example, [6], [13] and [14]), we first prove the following result: Theorem 1. Let
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| M (k ∈ N ). If a sequence X = (xk ) is 2 -statistically convergent to L, then it is 2 (stat)-summable to L, but not conversely. Proof. Since X = (xk ) is 2 -statistically convergent to L, it means that
lim
n→∞
1
λn − λn−1
|{k λn − λn−1 : |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| }| = 0.
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Let us set
K = {k λn − λn−1 : |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| } and
Kc = {k : λn − λn−1
and
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| < }.
We then find that
n 1 | − L| = (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L λn − λn−1 k=0 n 1 (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L λn − λn−1 k=1 (k∈K ) n 1 + (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L λn − λn−1 k=1 (k∈Kc ) 2
n 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| λn − λn−1 k=1
(k∈K )
+
n 1 λn − λn−1
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L|
k=1 (k∈Kc )
1 · M · |K| + λn − λn−1
→0+ ·1=
n 1 λn − λn−1 c k∈K
( n → ∞ ),
which implies that the sequence (xn ) is 2 -summable to L. That is, the sequence X = (xn ) is 2 -summable to L in the ordinary sense. This implies that the sequence X = (xn ) is 2 (stat)-statistically summable to L. To prove that the converse is not true, we construct the following example: Example 1. Let us consider the case when λn = n2 . Then
2 (x ) =
n 1 1 [n2 xn − (n − 1 )2 xn−1 ]. (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) = λn − λn−1 2n − 1 k=0
We now take
xk =
⎧ 1 ⎪ ⎪ ⎪ ⎨ m3+α 1
− 4 ⎪ ⎪ m ⎪ ⎩ 0
(α > 0; k = m2 − m, . . . , m2 − 1 ) (k = m2 ; m ∈ N \ {1} ) (otherwise).
Under these conditions, we get
lim 2 (x ) = 0
n→∞
and hence that
stat limn→∞ 2 (x ) = 0, that is, that the sequence X = (xn ) is 2 (stat)-summable to 0. On the other hand, since the sequence (m2 )m∈N\{1} is statistically convergent to 0, it is clear that
stat lim infn→∞ xn = 0 and
stat lim supn→∞ xn = 1.
Thus the sequence X = (xn ) is not statistically convergent, nor it is 2 -statistically convergent. Propositions 1, 2 and 3 below provide conditions under which 2 -statistical convergence implies 2r -summability and vice versa.
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Proposition 1. Let us suppose that the sequence X = (xn ) is 2r -summable to L. If
0 < r < 1 and 0 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| < 1 or
1 r < ∞ and 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| < ∞, then the sequence X = (xn ) is 2 -statistically convergent to L. Proof. Let us suppose that the sequence X = (xn ) is 2r -summable to L. Under the above conditions, we have
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L|r |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| in Case (1) and (2) of Proposition 1. Furthermore, we have
1 |K| = λn − λn−1
n 1 · (λn − λn−1 )
k=1 (k∈K )
n 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| · (λn − λn−1 ) k=1
(k∈K )
n 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| · (λn − λn−1 ) k=1
1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L|r → 0 · (λn − λn−1 )
( n → ∞ ).
k=1
Hence, the sequence X = (xn ) is 2 -statistically convergent to L. Proposition 2. Suppose that the sequence X = (xn ) is
2 -statistically
convergent to L and let
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| M (k ∈ N ). If
0 < r < 1 and 1 M < ∞ or
1 r < ∞ and 0 M < 1, then the sequence X = (xn ) is 2r -summable to L. Proposition 3. If X = (xk ) → L is statistically convergent, then it is 2 -statistically convergent to L, but not conversely. Furthermore, if (
λn −λn−1 n
) is a bounded sequence, then statistical convergence is equivalent to 2 -statistical convergence.
Theorem 2. A sequence X = (xn ) is 2 (stat)-summable to L if and only if there exists a set K = {r1 < r2 < · · · < rn < · · · } such that
δ (K ) = 1 and
lim 2 (xrn ) = L,
n→∞
where δ (K) denotes the natural density of the set K ⊂ N. Proof. Let us suppose that there exists a set K = {r1 < r2 < · · · < rn < · · · } ⊂ N such that
δ (K ) = 1 and
lim 2 (xrn ) = L.
n→∞
Then there is a positive integer n0 ∈ N such that, for n > n0 , we have rn 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| < . λn − λn−1 k=0
Put
K = {n : n ∈∈ N
and
|2 (xrn ) − L| }
and
K = {rn0 +1 , rn0 +2 , . . .}. Then
δ (K ) = 1 and K ⊂ N \ K ,
(1)
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679
which implies that
δ (K ) = 0. Hence, the sequence x = (xk ) is 2 (stat)-summable to L. Conversely, let the sequence x = (xk ) be 2 (stat)-summable to L. For r ∈ N, put
Kq =
and
Mq =
j : j ∈∈ N
j: j∈N
|2 (xr j ) − L|
and
|2 (xr j ) − L| <
and
1 q
1 . q
Then
δ (Kq ) = 0 and M1 ⊇ M2 ⊇ · · · ⊇ Mi ⊇ Mi+1 · · ·
(2)
and
δ (Mq ) = 1 (q ∈ N ).
(3)
We now have to show that for j ∈ Mq , (xk j) is
exists > 0 such that
2 -summable
to L. Suppose that (xk j) is not
2 -summable
to L. Therefore, there
|2 (xr j) − L| for infinitely many terms. Let
M = { j : j ∈ N
and
|2 (xr j) − L| < }
and
>
1 q
( q ∈ N ).
Then
δ (M ) = 0
(4)
and, by (2), Mq ⊂ M . Hence δ (Mq ) = 0, which contradicts (3) Therefore, (xk j) is
2 -summable
to L. This completes the proof of
Theorem 2.
3. Korovkin type theorems In this section, in order to investigate several Korovkin type theorems, we first introduce some notation related to the function spaces. By F[a, b] we denote the linear space of all real-valued functions defined on the closed interval [a, b]. We also use C[a, b] to denote the space of all bounded and continuous functions defined on [a, b]. It is a known fact that C[a, b] is a Banach space equipped with the following norm:
|| f ||∞ = sup | f (x )|
f ∈ C[a, b] .
x∈[a,b]
The classical Korovkin first theorem is recalled here as follows (see [8,9]; see also [1,15]): Theorem 3. Let (Bn ) be a sequence of positive linear operators from C[0, 1] into F[0, 1]. Then
lim
n→∞
||Bn ( f, x ) − f (x )||∞ = 0
for all f ∈ C[0, 1] if and only if
lim
n→∞
||Bn ( fi , x ) − fi (x )||∞ = 0 (i ∈ {0, 1, 2} ),
where
f0 (x ) = 1,
f 1 (x ) = x
and
f 2 ( x ) = x2 .
Remark 1. For a sequence (Bn ) of positive linear operators from [a, b] → [a, b], we say that the operator Bn is positive if Bn (f; x) ࣛ 0 whenever f(x) ࣛ 0 for x ∈ [a, b]. The Korovkin type theorems and their variants, extensions and generalizations have been investigated by several authors in many different ways and in many different settings such as (for example) function spaces, abstract Banach lattices, Banach algebras, and so on. This theory is useful in real analysis, functional analysis, harmonic analysis, and several other areas. For more
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results related to the Korovkin type theorems, one may see (for example) [4,5,10–12,16]. In this section, we extend the results of Boyanov and Veselinov [2] by using the notion of the 2 -summability method which we have defined above. Let C(I) be the Banach space with the uniform norm ||.||∞ of all real two-dimensional continuous functions on the interval I = [0, ∞ ) such that limx → ∞ f(x) is finite. Suppose also that Bn : C(I) → C(I). We write Bn (f; x) for Bn (f(s); x). Theorem 4. Let (Bk ) be a sequence of positive linear operators from C(I) into C(I). Then, for all f ∈ C(I)
2 (stat ) lim ||Bk ( f ; x ) − f (x )||∞ = 0 n→∞
(5)
if and only if
2 (stat ) lim ||Bk ( f ; x ) − 1||∞ = 0,
(6)
2 (stat ) lim ||Bk ( f ; x ) − e−x ||∞ = 0
(7)
2 (stat ) lim ||Bk ( f ; x ) − e−2x ||∞ = 0.
(8)
n→∞
n→∞
and n→∞
Proof. Let us suppose that the first assertion (5) is true. Since the functions 1, e−x and e−2x are continuous, the remaining assertions (6), (7) and (8) follow immediately from the first assertion (5). Now we will prove the converse, that is, that the assertions (6, 7 and 8) are valid, and also that the assertion (5) is valid, too. Let f ∈ C(I). Then there exists a constant κ > 0 such that |f(x)| ࣚ κ for all x ∈ (I). Therefore, we have
| f (t ) − f (x )| 2κ (x ∈ I ).
(9)
For every given > 0, there exists a δ > 0 such that
| f (t ) − f (x )| whenever |e−t − e−x | < δ (x ∈ I ).
(10)
Let us set
ψ ≡ ψ (t, x ) = (e−t − e−x )2 . If |t − x| δ, then we have
2κ
| f (t ) − f (x )|
δ2
ψ (t, x ).
(11)
Now, from the relations (9) to (11), we get
| f (t ) − f (x )| < +
2κ
δ2
ψ (t, x )
and
− −
2κ
δ2
ψ (t, x ) < f (t ) − f (x ) <
2κ
δ2
ψ (t, x ) + ,
respectively. Since Bk (1, x) is monotone and linear, by applying the operator Bk (1, x) in this last inequality, we see that the following inequalities:
Bk (1, x ) − −
2κ
ψ < Bk (1, x )[ f (t ) − f (x )] < Bk (1, x ) 2 ψ + δ δ
2κ 2
imply that
− Bk (1, x ) −
2κ
δ2
Bk
2κ
ψ (t ), x < Bk ( f, x ) − f (x )Bk (1, x ) < 2 Bk ψ (t ), x + Bk (1, x ). δ
(12)
On the other hand, we have
Bk ( f, x ) − f (x ) = Bk ( f, x ) − f (x )Bk (1, x ) + f (x )[Bk (1, x ) − 1].
(13)
From the relations (12) and (13), we have
Bk ( f, x ) − f (x ) <
2κ
δ2
Bk
ψ (t ), x + Bk (1, x ) + f (x )[Bk (1, x ) − 1].
Let us now estimate the following expression:
Bk
ψ (t ), x = Bk (e−x − e−t )2 , x = Bk (e−2x − 2e−x e−t + e−2t ), x = Bk (1, x ) − 2e−x Bk (e−t , x ) + Bk (e−2t , x ),
(14)
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681
which, in conjunction with (14), yields
Bk ( f, x ) − f (x ) <
2κ
e−2x [Bk (1, x ) − 1] − 2e−x [Bk (e−t , x ) − e−x ]
δ2
+ [Bk (e−2t , x ) − e−2x ] + Bk (1, x ) + f (x )[Bk (1, x ) − 1] = + [Bk (1, x ) − 1] + f (x )[Bk (1, x ) − 1] 2κ
+
δ2
e−2x [Bk (1, x ) − 1] − 2e−x [Bk (e−t , x ) − e−x ]
+ [Bk (e−2t , x ) − e−2x ] . Therefore, we get
2κ |Bk ( f, x ) − f (x )| + + κ + 2 |Bk (1, x ) − 1| δ 4κ −t + 2 |Bk (e , x ) − e−x | δ 2κ + 2 |Bk (e−2t , x ) − e−2x |. δ
Now, taking the supx∈I in the above relation, we find that
||Bk ( f, x ) − f (x )||C (I ) + M ||Bk (1, x ) − 1||C (I ) + ||Bk (e−t , x ) − e−x ||C (I )
(15)
+ ||Bk (e−2t , x ) − e−2x ||C (I ) , where
M = max
+κ +
(16)
2κ 4κ , 2 . 2
δ
δ
We now replace Bk (., x) by
2 =
n 1 (λk B(., x ) − 2λk−1 B(., x ) + λk−2 B(., x )) λn − λn−1 k=0
on both sides of the above inequality (15). For a given r > 0, we can choose 1 such that 1 < r. By defining the following sets:
D = {k : k ∈ N and
and
Di = k : k ∈ N
and
||2 ( f, x ) − f (x )||C (I ) r} ||2 ( fi , x ) − fi (x )||C (I )
r − 1 3κ
(i = 0, 1, 2, 3 ) ,
we have
D ⊂ ∪2i=0 Di and their densities is satisfy the relation:
δ (D ) δ (D0 ) + δ (D1 ) + δ (D2 ). Finally, from the relations (6), (7) and (8) and the above estimation, we get
2 (stat ) lim ||2 ( f ; x ) − f (x )||∞ = 0, n→∞
which completes the proof of Theorem 4. Theorem 4 is valid also for test functions fi (x ) = xi for i ∈ {0, 1, 2} (see also remark 2 below). Remark 2. Let (Bk ) be a sequence of positive linear operators from C(I) into C(I). Then, for all f ∈ C(I),
2 (stat ) lim ||Bk ( f ; x ) − f (x )||∞ = 0 n→∞
(17)
if and only if
2 (stat ) lim ||Bk ( f ; x ) − 1||∞ = 0,
(18)
2 (stat ) lim ||Bk ( f ; x ) − x||∞ = 0
(19)
n→∞
n→∞
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and
2 (stat ) lim ||Bk ( f ; x ) − x2 ||∞ = 0. n→∞
(20)
We next give an example which shows that our result asserted by Theorem 4 is an extension of that given in [2]. Example 2. We will consider the following type of classical Baskakov operators (see [18]):
Vn ( f, x ) =
∞ k=0
f
k n
n+k−1 k x · (1 + x )−(n+k) , k
where x ∈ [0, ∞). (see [18]). On the other hand, let Ln : C(I) → C(I) define the following type of operators:
Ln ( f, x ) = (1 + xn )Vn ( f, x ), where the sequence X = (xn ) is defined by Example 1. We note that this sequence is 2 -statistically summable to 0, but neither convergent nor statistically convergent. Then we get the following estimation:
Ln (1, x ) = 1, 1
Ln (e−s , x ) = (1 + x − xe− n )−n and 2
Ln (e−2s , x ) = (1 + x − xe− n )−n . Hence, by Theorem 4, it follows that
2 (stat ) lim ||Ln ( f ; x ) − f (x )||∞ = 0. n→∞
Thus, from the above estimations, the sequence (Ln ) does not satisfy the conditions of the theorem given by Boyanov and Veselinov (see [2]). Hence, our theorem is stronger than that of Boyanov and Veselinov (see [2]). 4. Rates of convergence In this section, we study the rate of the weighted statistical convergence of a sequence of positive linear operators 2 defined on C[a, b]. We begin by presenting the following definition: Definition 4. Let (an ) be any nonincreasing sequence of positive real numbers. We say that the sequence X = (xn ) is 2 statistically convergent to the number L with the rate of convergence o(an ) if, for every > 0,
lim
n→∞
1 |{k λn − λn−1 : |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| }| = 0. an (λn − λn−1 )
In this case, we write
xk − L = stat2 o(an ). Lemma 1. Let (an ) and (bn ) be two nonincreasing sequences of positive real numbers. Also let X = (xn ) and Y = (yn ) be two sequences such that
xn − L1 = stat2 o(an ) and
yn − L2 = stat2 o(bn ).
Then
(xn − L1 ) ± (yn − L2 ) = stat2 o(cn ),
α (xn − L ) = stat2 o(an ) (α ∈ R ) and
(xn − L1 )(yn − L2 ) = stat2 o(an bn ), where
cn = max {an , bn }. n∈N
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683
Proof. We only prove just the first assertion of Lemma 1; the other assertions can be proved similarly. For a given > 0, let us set
A1 = {k : k λn − λn−1
and
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) + (λk yk − 2λk−1 yk−1 + λk−2 yk−2 ) − (L1 + L2 )| },
and
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L1 |
and
|(λk yk − 2λk−1 yk−1 + λk−2 yk−2 ) − L2 |
A2 = k : k λn − λn−1 and
A3 = k : k λn − λn−1
2
2
.
Then we observe that A1 ⊂ A2 ∪ A3 . Moreover, since
cn = max {an , bn }, n∈N
we obtain
|A1 | |A2 | |A2 | + . (λn − λn−1 ) · cn (λn − λn−1 ) · cn (λn − λn−1 ) · cn
(21)
Now, by taking the limit as n → ∞ in (21) and using the hypothesis of Lemma 1, we conclude that
lim
n→∞
|A1 | = 0, (λn − λn−1 ) · cn
which proves the first assertion of Lemma 1. We now recall that the modulus of continuity for a function f (x ) ∈ C2π (R ) is defined as follows:
ω ( f, δ ) = sup | f (x + h ) − f (x )|. |h|<δ
We consider the following two cases: (A) If |x − y| < δ, then
| f (x ) − f (y )| ω ( f, δ ) and (B) If |x − y| > δ, then
| f (x ) − f (y )| ω ( f, δ ) ω ( f, δ ) ·
|x − y| . δ
From these last two relations, we find for any value of |x − y| that
| f (x ) − f (y )| ω ( f, δ )
|x − y| 2
+1 .
We state and prove the following result. Theorem 5. Let (Bn ) be a sequence of positive linear operators from C[a, b] into C[a, b]. Suppose that
||Bn (1, x ) − 1||∞ = stat2 o(an ) and
ω ( f, λn ) = stat2 o(bn ), where
λn =
Bn ( ψ , x )
and
ψ ≡ ψ (t, x ) = (e−t − e−x )2 .
Then, for all f ∈ C[a, b] and x ∈ [a, b], we have
||Bn ( f, x ) − f (x )||∞ = stat2 − o(cn ), where
cn = max {an , bn }. n∈N
(22)
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N.L. Braha et al. / Applied Mathematics and Computation 266 (2015) 675–686
Proof. Let f ∈ C[a, b] and x ∈ [a, b]. From the relations (13) and (22), we get the following estimation:
|Bn ( f, x ) − f (x )| |Bn (| f (y ) − f (x )|, x )| + | f (x )| · |Bn (1, x ) − 1| |x − y| Bn + 1, x ω ( f, δ ) + | f (x )| · |Bn (1, x ) − 1| δ
M Bn 1 + (e−t − e−x )2 , x ω ( f, δ ) + | f (x )| · |Bn (1, x ) − 1| δ
M Bn (1, x ) + Bn (ψ , x ) ω ( f, δ ) + | f (x )| · |Bn (1, x ) − 1|. δ If we put
δ=
λn 2 M
=
Bn ( ψ , x ) M
in the last relation, we obtain
||Bn ( f, x ) − f (x )||∞ || f ||∞ ||Bn (1, x ) − 1||∞ + ω ( f, λn ) + ω ( f, λn )||Bn (1, x ) − 1||∞ C {||Bn (1, x ) − 1||∞ + ω ( f, λn ) + ω ( f, λn )||Bn (1, x ) − 1||∞ }, where
C = max
|| f ||∞ , 1 .
Now, upon replacing Bk (., x) by
2 =
n 1 (λk Bk (., x ) − 2λk−1 Bk−1 (., x ) + λk−2 Bk−2 (., x )), λn − λn−1 k=0
we get
||2 ( f, x ) − f (x )||∞ C ||2 (1, x ) − 1||∞ + ω ( f, λn ) + ω ( f, λn )||2 (1, x ) − 1||∞ .
The proof of Theorem 5 now follows readily from the hypotheses and Lemma 1. 5. Voronovskaya type theorems In this section, we show that the positive linear operators Ln defined in Example 2 satisfy a Voronovskaja type property in the
2 (stat) sense. We first prove the following lemma: Lemma 2. For x ∈ [0, 1] and (y ) = y − x,
n2 Ln ( 4 ) ∼ 3x2 (1 + x )2 (2 (stat )) on
[0, 1].
Proof. After some calculations, we get
x(1 + x )(6x2 + 6x + 1 ) 3x2 (1 + x )2 n Ln ( ) = n (1 + xn ) + , n2 n3 2
4
2
which shows that
2 n Ln ( 4 ) − 3x2 (1 + x )2 2 x ( 1 + x )( 6x + 6x + 1 ) 2 2 2 2 − 3x (1 + x ) = (1 + xn )3x (1 + x ) + (1 + xn ) n
2 x(1 + x )(6x2 + 6x + 1 ) 3x2 (1 + x )2 |(1 + xn ) − 1| + (1 + xn ) → 0 (stat ) on [0, 1]. n
This evidently completes proof of Lemma 2. The following Voronovskaya type theorem for operators Ln given in Example 2 holds true. Theorem 6. For every f ∈ C[0, 1] such that f , f ∈ C[0, 1], then
n Ln ( f ) − f ∼ on [0, 1].
1 (x + x2 ) f (x ) 2
2 (stat )
(23)
N.L. Braha et al. / Applied Mathematics and Computation 266 (2015) 675–686
685
Proof. Let us suppose that f , f ∈ C[0, 1] and x ∈ [0, 1]. Define
f (y ) − f (x ) − (y − x ) f (x ) − 12 (y − x )2 f (x ) ( y − x )2 0
ψx (y ) =
(x = y ) ( x = y ).
Then ψx (x ) = 0 and ψ x ∈ C[0, 1]. By Taylor’s formula, we get
f (y ) = f (x ) + (y − x ) f (x ) +
1 (y − x ) f (x ) + (y − x )2 ψx (y ). 2
(24)
Knowing that
Ln (1, x ) = (1 + xn ); Ln (y − x ), x = 0 and
Ln (y − x )2 , x = (1 + xn )
x (1 + x ) , n
and after operating on both sides of the relation (24) by the operator Ln , we obtain
f (x ) x(1 + x ) (1 + xn ) + (1 + xn )Vn ( 2 ψx , x ), 2 n
Ln ( f ) = f ( x ) + xn f ( x ) + which yields
1 n[Ln ( f ) − f (x )] − (x + x2 ) f (x ) nxn | f (x )| + xn | f (x )| + n(1 + xn )Vn ( 2 ψx , x ) 2
and
1 n[Ln ( f ) − f (x )] − 2 (x + x2 ) f (x ) nxn M + n(1 + xn )Vn ( 2 ψx , x ),
(25)
where
(y ) = y − x and M = || f ||C[0,1] + || f ||C[0,1] . After application of the Cuachy–Schwarz inequality in the second term of the right-hand side of the relation (25), we find that
nVn ( 2 ψx , x ) [n2Vn ( 4 , x )] 2 · [Vn (ψx , x )] 2 . 1
1
(26)
2 Also, by putting ηx (y ) = ψx (y ) , we see that ηx (x ) = 0 and ηx ( · ) ∈ C[0, 1]. Clearly, from Remark 2, it follows that
Ln (ηx ) → 0 2 (stat )
on
[0, 1].
(27)
Now, from the relations (26) and (27) and Lemma 2, we have
Ln ( 2 ψx , x ) → 0 2 (stat )
on [0, 1].
(28)
For a given > 0, we define the following sets:
An (x, ) = k : k λn − λn−1
A1,n (x, ) = k : k λn − λn−1 and
1 k(Vk ( f, x ) − f (x )) − 2 (x + x2 ) f (x ) , and |kxn |
and
A2,n (x, ) = k : k λn − λn−1
2M
and
|kVn ( 2 ψx , x )|
. 2
From these last relations, we obtain
An (x, ) A1,n (x, ) + A2,n (x, ) and
2 (An ·, ) 2 (A1,n ·, ) + 2 A2,n (·, ) .
(29)
Moreover, by the definition of the sequence X = (xn ), we get
nxn → 0
2 (stat )
on
[0, 1].
(30)
Now, from the relations (28) and (30), the right-hand side of the relation (29) tends to zero as n → ∞. Therefore, we have
stat limn→∞ 2 An (·, ) = 0, which proves that
n Ln ( f ) − f ∼
1 (x + x2 ) f (x ) 2 (stat ) on 2
[0, 1].
We have thus completed our proof of Theorem 6. Remark 3. Since the sequence X = (xn ) given in Example 1 is not statistically convergent, we conclude that the operators (Ln ) defined in Example 2 do not satisfy a Voronovskaya type theorem in the usual sense.
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References [1] F. Altomare, Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory 5 (2010) 92–164. [2] B.D. Boyanov, V.M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 14 (62) (1970) 9–13. [3] N.L. Braha, T. Mansour, On λ2 -strong convergence of numerical sequences and fourier series, Acta Math. Hungar. 141 (2013) 113–126. [4] N.L. Braha, H.M. Srivastava, S.A. Mohiuddine, A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la vallée poussin mean, Appl. Math. Comput. 228 (2014) 162–169. [5] O.H.H. Edely, S.A. Mohiuddine, A.K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett. 23 (2010) 1382–1387. [6] E. Erkus, ¸ O. Duman, H.M. Srivastava, Statistical approximation of certain positive linear operators constructed by means of the Chan–Chyan–Srivastava polynomials, Appl. Math. Comput. 182 (2006) 213–222. [7] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244. [8] P.P. Korovkin, Convergence of linear positive operators in the spaces of continuous functions, Dokl. Akad. Nauk. SSSR (N.S.) 90 (1953) 961–964. (in Russian) [9] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corporation, Delhi, 1960. [10] S.A. Mohiuddine, A. Alotaibi, M. Mursaleen, Statistical summability (c, 1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012) 1–8. Article ID 2012:172 [11] M. Mursaleen, A. Alotaibi, Statistical lacunary summability and a Korovkin type approximation theorem, Ann. Univ. Ferrara 57 (2011) 373–381. [12] M. Mursaleen, V. Karakaya, M. Erturk, F. Gursoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput. 218 (2012) 9132–9137. [13] M. Mursaleen, A. Khan, H.M. Srivastava, K.S. Nisar, Operators constructed by means of q-Lagrange polynomials and a-statistical approximation, Appl. Math. Comput. 219 (2013) 6911–6918. [14] M.A. Özarslan, O. Duman, H.M. Srivastava, Statistical approximation results for Kantorovich-type operators involving some special polynomials, Math. Comput. Modelling 48 (2008) 388–401. [15] R. P˘alt˘anea, Approximation Theory Using Positive Linear Operators, Birkhäuser, Boston, Basel and Berlin, 2004. [16] H.M. Srivastava, M. Mursaleen, A. Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Modelling 55 (2012) 2040–2051. [17] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951) 73–74. [18] J.J. Wang, Z.B. Xu, Approximation with Jacobi weights by Baskakov operators, Taiwanese J. Math. 13 (2009) 157–168.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
2 -Weighted statistical convergence and Korovkin and Voronovskaya type theorems Naim L. Braha a, Valdete Loku a, H.M. Srivastava b,∗ a Department of Computer Sciences and Applied Mathematics, College Vizioni Për Arsim, Street Ahmet Kaciku Nr. 3, Kati i dyte, Ferizaj 70000, Republic of Kosova b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
a r t i c l e
i n f o
a b s t r a c t In this paper, we propose to introduce a new 2 -weighted statistical convergence. Based upon this definition, we prove some Korovkin type theorems. We also find the rate of the convergence for this kind of weighted statistical convergence and derive some Voronovskaya type theorems.
MSC: Primary 40C05 40G15 41A36 Secondary 46A35 46A45 46B45
© 2015 Elsevier Inc. All rights reserved.
Keywords: Statistical convergence and statistical summability 2 -Weighted statistical convergence and Korovkin type theorems Rate of convergence and Voronovskaya type theorems Positive linear operators Bounded and continuous functions and modulus of continuity Nonincreasing and nondecreasing functions
1. Introduction, definitions and preliminaries We denote by N the set of all natural numbers. We also let K ⊂ N and suppose that
Kn := {k : k n and
k ∈ K}.
Then the natural density of K is defined by
d (K ) := lim
n→∞
|Kn | n
= lim
n→∞
1 |{k : k n and n
k ∈ K}|
if the limit exists. Here, and in other similar situations, by || we denote the number of elements in the enclosed set . A given sequence X = (xn ) is said to be statistically convergent to L if, for each > 0, the set
K := {k : k ∈ N ∗
and
|xk − L| }
Corresponding author. Tel.: +1 250 472 5313; fax: +1 250 721 8962. E-mail addresses: [email protected] (N.L. Braha), [email protected] (V. Loku), [email protected] (H.M. Srivastava).
http://dx.doi.org/10.1016/j.amc.2015.05.108 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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has natural density zero (see [7] and [17]), that is, for each > 0, we have
lim
n→∞
|K | n
= lim
n→∞
1 |{k : k n and n
|xk − L| }| = 0.
In this case, we write
L = stat limn→∞ xn . We note here that every convergent sequence is statistically convergent, but not conversely. be a strictly increasing sequence of positive real numbers which tend to infinity as n → ∞, that is, let Let = (λn )∞ n=0
0 < λ0 < λ1 < · · · < λn < · · · ,
lim
n→∞
λn = ∞ and 2 (λn ) 0,
where the first difference (λk ) and the second difference 2 (λk ) are defined by
λk = λk − λk−1 (λ−1 = λ−2 = 0 ) and
2 (λk ) = ((λk )) = λk − 2λk−1 + λk−2 (λ−1 = λ−2 = 0 ), respectively. Let X = (xk ) be a sequence of complex numbers such that x−1 = x−2 = 0. We then denote by 2n (x ) the following sum (see [3]):
2n (x ) =
n 1 (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ). λn − λn−1 k=0
Definition 1. The sequence (xn ) is said to be 2 -summable to L if
lim 2n (x ) = L.
n→∞
We also say that the sequence (xn ) is statistically summable to L by the weighted method determined by the sequence 2n (x ) if
stat limn→∞ 2n (x ) = L. We denote by 2 (stat) the set of all sequences which are 2 -statistically summable. Definition 2. We say that the sequence (xn ) is weighted 2 -statistically convergent to L if, for every > 0,
lim
n→∞
1
λn − λn−1
|{k λn − λn−1 : |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| }| = 0.
In this case, we write
L = stat2 limn→∞ xn . Definition 3. A sequence X = (xn ) is said to be strongly 2r -summable (0 < r < ∞) to the limit L if
lim
n→∞
n 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L|r = 0. λn − λn−1 k=1
We write it as xk → 2r . In this case, L is called strongly 2r limit of the sequence X = (xn ). Based upon the above definition of the 2 -weighted statistical convergence, we propose to prove here some Korovkin type theorems. We also find the rate of the convergence for this kind of weighted statistical convergence and derive some Voronovskaya type theorems. 2. Relations involving the 2 -statistically summable and convergent sequences and strongly 2r -summability sequences Motivated essentially by several recent works (see, for example, [6], [13] and [14]), we first prove the following result: Theorem 1. Let
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| M (k ∈ N ). If a sequence X = (xk ) is 2 -statistically convergent to L, then it is 2 (stat)-summable to L, but not conversely. Proof. Since X = (xk ) is 2 -statistically convergent to L, it means that
lim
n→∞
1
λn − λn−1
|{k λn − λn−1 : |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| }| = 0.
N.L. Braha et al. / Applied Mathematics and Computation 266 (2015) 675–686
677
Let us set
K = {k λn − λn−1 : |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| } and
Kc = {k : λn − λn−1
and
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| < }.
We then find that
n 1 | − L| = (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L λn − λn−1 k=0 n 1 (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L λn − λn−1 k=1 (k∈K ) n 1 + (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L λn − λn−1 k=1 (k∈Kc ) 2
n 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| λn − λn−1 k=1
(k∈K )
+
n 1 λn − λn−1
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L|
k=1 (k∈Kc )
1 · M · |K| + λn − λn−1
→0+ ·1=
n 1 λn − λn−1 c k∈K
( n → ∞ ),
which implies that the sequence (xn ) is 2 -summable to L. That is, the sequence X = (xn ) is 2 -summable to L in the ordinary sense. This implies that the sequence X = (xn ) is 2 (stat)-statistically summable to L. To prove that the converse is not true, we construct the following example: Example 1. Let us consider the case when λn = n2 . Then
2 (x ) =
n 1 1 [n2 xn − (n − 1 )2 xn−1 ]. (λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) = λn − λn−1 2n − 1 k=0
We now take
xk =
⎧ 1 ⎪ ⎪ ⎪ ⎨ m3+α 1
− 4 ⎪ ⎪ m ⎪ ⎩ 0
(α > 0; k = m2 − m, . . . , m2 − 1 ) (k = m2 ; m ∈ N \ {1} ) (otherwise).
Under these conditions, we get
lim 2 (x ) = 0
n→∞
and hence that
stat limn→∞ 2 (x ) = 0, that is, that the sequence X = (xn ) is 2 (stat)-summable to 0. On the other hand, since the sequence (m2 )m∈N\{1} is statistically convergent to 0, it is clear that
stat lim infn→∞ xn = 0 and
stat lim supn→∞ xn = 1.
Thus the sequence X = (xn ) is not statistically convergent, nor it is 2 -statistically convergent. Propositions 1, 2 and 3 below provide conditions under which 2 -statistical convergence implies 2r -summability and vice versa.
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N.L. Braha et al. / Applied Mathematics and Computation 266 (2015) 675–686
Proposition 1. Let us suppose that the sequence X = (xn ) is 2r -summable to L. If
0 < r < 1 and 0 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| < 1 or
1 r < ∞ and 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| < ∞, then the sequence X = (xn ) is 2 -statistically convergent to L. Proof. Let us suppose that the sequence X = (xn ) is 2r -summable to L. Under the above conditions, we have
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L|r |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| in Case (1) and (2) of Proposition 1. Furthermore, we have
1 |K| = λn − λn−1
n 1 · (λn − λn−1 )
k=1 (k∈K )
n 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| · (λn − λn−1 ) k=1
(k∈K )
n 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| · (λn − λn−1 ) k=1
1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L|r → 0 · (λn − λn−1 )
( n → ∞ ).
k=1
Hence, the sequence X = (xn ) is 2 -statistically convergent to L. Proposition 2. Suppose that the sequence X = (xn ) is
2 -statistically
convergent to L and let
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| M (k ∈ N ). If
0 < r < 1 and 1 M < ∞ or
1 r < ∞ and 0 M < 1, then the sequence X = (xn ) is 2r -summable to L. Proposition 3. If X = (xk ) → L is statistically convergent, then it is 2 -statistically convergent to L, but not conversely. Furthermore, if (
λn −λn−1 n
) is a bounded sequence, then statistical convergence is equivalent to 2 -statistical convergence.
Theorem 2. A sequence X = (xn ) is 2 (stat)-summable to L if and only if there exists a set K = {r1 < r2 < · · · < rn < · · · } such that
δ (K ) = 1 and
lim 2 (xrn ) = L,
n→∞
where δ (K) denotes the natural density of the set K ⊂ N. Proof. Let us suppose that there exists a set K = {r1 < r2 < · · · < rn < · · · } ⊂ N such that
δ (K ) = 1 and
lim 2 (xrn ) = L.
n→∞
Then there is a positive integer n0 ∈ N such that, for n > n0 , we have rn 1 |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| < . λn − λn−1 k=0
Put
K = {n : n ∈∈ N
and
|2 (xrn ) − L| }
and
K = {rn0 +1 , rn0 +2 , . . .}. Then
δ (K ) = 1 and K ⊂ N \ K ,
(1)
N.L. Braha et al. / Applied Mathematics and Computation 266 (2015) 675–686
679
which implies that
δ (K ) = 0. Hence, the sequence x = (xk ) is 2 (stat)-summable to L. Conversely, let the sequence x = (xk ) be 2 (stat)-summable to L. For r ∈ N, put
Kq =
and
Mq =
j : j ∈∈ N
j: j∈N
|2 (xr j ) − L|
and
|2 (xr j ) − L| <
and
1 q
1 . q
Then
δ (Kq ) = 0 and M1 ⊇ M2 ⊇ · · · ⊇ Mi ⊇ Mi+1 · · ·
(2)
and
δ (Mq ) = 1 (q ∈ N ).
(3)
We now have to show that for j ∈ Mq , (xk j) is
exists > 0 such that
2 -summable
to L. Suppose that (xk j) is not
2 -summable
to L. Therefore, there
|2 (xr j) − L| for infinitely many terms. Let
M = { j : j ∈ N
and
|2 (xr j) − L| < }
and
>
1 q
( q ∈ N ).
Then
δ (M ) = 0
(4)
and, by (2), Mq ⊂ M . Hence δ (Mq ) = 0, which contradicts (3) Therefore, (xk j) is
2 -summable
to L. This completes the proof of
Theorem 2.
3. Korovkin type theorems In this section, in order to investigate several Korovkin type theorems, we first introduce some notation related to the function spaces. By F[a, b] we denote the linear space of all real-valued functions defined on the closed interval [a, b]. We also use C[a, b] to denote the space of all bounded and continuous functions defined on [a, b]. It is a known fact that C[a, b] is a Banach space equipped with the following norm:
|| f ||∞ = sup | f (x )|
f ∈ C[a, b] .
x∈[a,b]
The classical Korovkin first theorem is recalled here as follows (see [8,9]; see also [1,15]): Theorem 3. Let (Bn ) be a sequence of positive linear operators from C[0, 1] into F[0, 1]. Then
lim
n→∞
||Bn ( f, x ) − f (x )||∞ = 0
for all f ∈ C[0, 1] if and only if
lim
n→∞
||Bn ( fi , x ) − fi (x )||∞ = 0 (i ∈ {0, 1, 2} ),
where
f0 (x ) = 1,
f 1 (x ) = x
and
f 2 ( x ) = x2 .
Remark 1. For a sequence (Bn ) of positive linear operators from [a, b] → [a, b], we say that the operator Bn is positive if Bn (f; x) ࣛ 0 whenever f(x) ࣛ 0 for x ∈ [a, b]. The Korovkin type theorems and their variants, extensions and generalizations have been investigated by several authors in many different ways and in many different settings such as (for example) function spaces, abstract Banach lattices, Banach algebras, and so on. This theory is useful in real analysis, functional analysis, harmonic analysis, and several other areas. For more
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results related to the Korovkin type theorems, one may see (for example) [4,5,10–12,16]. In this section, we extend the results of Boyanov and Veselinov [2] by using the notion of the 2 -summability method which we have defined above. Let C(I) be the Banach space with the uniform norm ||.||∞ of all real two-dimensional continuous functions on the interval I = [0, ∞ ) such that limx → ∞ f(x) is finite. Suppose also that Bn : C(I) → C(I). We write Bn (f; x) for Bn (f(s); x). Theorem 4. Let (Bk ) be a sequence of positive linear operators from C(I) into C(I). Then, for all f ∈ C(I)
2 (stat ) lim ||Bk ( f ; x ) − f (x )||∞ = 0 n→∞
(5)
if and only if
2 (stat ) lim ||Bk ( f ; x ) − 1||∞ = 0,
(6)
2 (stat ) lim ||Bk ( f ; x ) − e−x ||∞ = 0
(7)
2 (stat ) lim ||Bk ( f ; x ) − e−2x ||∞ = 0.
(8)
n→∞
n→∞
and n→∞
Proof. Let us suppose that the first assertion (5) is true. Since the functions 1, e−x and e−2x are continuous, the remaining assertions (6), (7) and (8) follow immediately from the first assertion (5). Now we will prove the converse, that is, that the assertions (6, 7 and 8) are valid, and also that the assertion (5) is valid, too. Let f ∈ C(I). Then there exists a constant κ > 0 such that |f(x)| ࣚ κ for all x ∈ (I). Therefore, we have
| f (t ) − f (x )| 2κ (x ∈ I ).
(9)
For every given > 0, there exists a δ > 0 such that
| f (t ) − f (x )| whenever |e−t − e−x | < δ (x ∈ I ).
(10)
Let us set
ψ ≡ ψ (t, x ) = (e−t − e−x )2 . If |t − x| δ, then we have
2κ
| f (t ) − f (x )|
δ2
ψ (t, x ).
(11)
Now, from the relations (9) to (11), we get
| f (t ) − f (x )| < +
2κ
δ2
ψ (t, x )
and
− −
2κ
δ2
ψ (t, x ) < f (t ) − f (x ) <
2κ
δ2
ψ (t, x ) + ,
respectively. Since Bk (1, x) is monotone and linear, by applying the operator Bk (1, x) in this last inequality, we see that the following inequalities:
Bk (1, x ) − −
2κ
ψ < Bk (1, x )[ f (t ) − f (x )] < Bk (1, x ) 2 ψ + δ δ
2κ 2
imply that
− Bk (1, x ) −
2κ
δ2
Bk
2κ
ψ (t ), x < Bk ( f, x ) − f (x )Bk (1, x ) < 2 Bk ψ (t ), x + Bk (1, x ). δ
(12)
On the other hand, we have
Bk ( f, x ) − f (x ) = Bk ( f, x ) − f (x )Bk (1, x ) + f (x )[Bk (1, x ) − 1].
(13)
From the relations (12) and (13), we have
Bk ( f, x ) − f (x ) <
2κ
δ2
Bk
ψ (t ), x + Bk (1, x ) + f (x )[Bk (1, x ) − 1].
Let us now estimate the following expression:
Bk
ψ (t ), x = Bk (e−x − e−t )2 , x = Bk (e−2x − 2e−x e−t + e−2t ), x = Bk (1, x ) − 2e−x Bk (e−t , x ) + Bk (e−2t , x ),
(14)
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681
which, in conjunction with (14), yields
Bk ( f, x ) − f (x ) <
2κ
e−2x [Bk (1, x ) − 1] − 2e−x [Bk (e−t , x ) − e−x ]
δ2
+ [Bk (e−2t , x ) − e−2x ] + Bk (1, x ) + f (x )[Bk (1, x ) − 1] = + [Bk (1, x ) − 1] + f (x )[Bk (1, x ) − 1] 2κ
+
δ2
e−2x [Bk (1, x ) − 1] − 2e−x [Bk (e−t , x ) − e−x ]
+ [Bk (e−2t , x ) − e−2x ] . Therefore, we get
2κ |Bk ( f, x ) − f (x )| + + κ + 2 |Bk (1, x ) − 1| δ 4κ −t + 2 |Bk (e , x ) − e−x | δ 2κ + 2 |Bk (e−2t , x ) − e−2x |. δ
Now, taking the supx∈I in the above relation, we find that
||Bk ( f, x ) − f (x )||C (I ) + M ||Bk (1, x ) − 1||C (I ) + ||Bk (e−t , x ) − e−x ||C (I )
(15)
+ ||Bk (e−2t , x ) − e−2x ||C (I ) , where
M = max
+κ +
(16)
2κ 4κ , 2 . 2
δ
δ
We now replace Bk (., x) by
2 =
n 1 (λk B(., x ) − 2λk−1 B(., x ) + λk−2 B(., x )) λn − λn−1 k=0
on both sides of the above inequality (15). For a given r > 0, we can choose 1 such that 1 < r. By defining the following sets:
D = {k : k ∈ N and
and
Di = k : k ∈ N
and
||2 ( f, x ) − f (x )||C (I ) r} ||2 ( fi , x ) − fi (x )||C (I )
r − 1 3κ
(i = 0, 1, 2, 3 ) ,
we have
D ⊂ ∪2i=0 Di and their densities is satisfy the relation:
δ (D ) δ (D0 ) + δ (D1 ) + δ (D2 ). Finally, from the relations (6), (7) and (8) and the above estimation, we get
2 (stat ) lim ||2 ( f ; x ) − f (x )||∞ = 0, n→∞
which completes the proof of Theorem 4. Theorem 4 is valid also for test functions fi (x ) = xi for i ∈ {0, 1, 2} (see also remark 2 below). Remark 2. Let (Bk ) be a sequence of positive linear operators from C(I) into C(I). Then, for all f ∈ C(I),
2 (stat ) lim ||Bk ( f ; x ) − f (x )||∞ = 0 n→∞
(17)
if and only if
2 (stat ) lim ||Bk ( f ; x ) − 1||∞ = 0,
(18)
2 (stat ) lim ||Bk ( f ; x ) − x||∞ = 0
(19)
n→∞
n→∞
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and
2 (stat ) lim ||Bk ( f ; x ) − x2 ||∞ = 0. n→∞
(20)
We next give an example which shows that our result asserted by Theorem 4 is an extension of that given in [2]. Example 2. We will consider the following type of classical Baskakov operators (see [18]):
Vn ( f, x ) =
∞ k=0
f
k n
n+k−1 k x · (1 + x )−(n+k) , k
where x ∈ [0, ∞). (see [18]). On the other hand, let Ln : C(I) → C(I) define the following type of operators:
Ln ( f, x ) = (1 + xn )Vn ( f, x ), where the sequence X = (xn ) is defined by Example 1. We note that this sequence is 2 -statistically summable to 0, but neither convergent nor statistically convergent. Then we get the following estimation:
Ln (1, x ) = 1, 1
Ln (e−s , x ) = (1 + x − xe− n )−n and 2
Ln (e−2s , x ) = (1 + x − xe− n )−n . Hence, by Theorem 4, it follows that
2 (stat ) lim ||Ln ( f ; x ) − f (x )||∞ = 0. n→∞
Thus, from the above estimations, the sequence (Ln ) does not satisfy the conditions of the theorem given by Boyanov and Veselinov (see [2]). Hence, our theorem is stronger than that of Boyanov and Veselinov (see [2]). 4. Rates of convergence In this section, we study the rate of the weighted statistical convergence of a sequence of positive linear operators 2 defined on C[a, b]. We begin by presenting the following definition: Definition 4. Let (an ) be any nonincreasing sequence of positive real numbers. We say that the sequence X = (xn ) is 2 statistically convergent to the number L with the rate of convergence o(an ) if, for every > 0,
lim
n→∞
1 |{k λn − λn−1 : |(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L| }| = 0. an (λn − λn−1 )
In this case, we write
xk − L = stat2 o(an ). Lemma 1. Let (an ) and (bn ) be two nonincreasing sequences of positive real numbers. Also let X = (xn ) and Y = (yn ) be two sequences such that
xn − L1 = stat2 o(an ) and
yn − L2 = stat2 o(bn ).
Then
(xn − L1 ) ± (yn − L2 ) = stat2 o(cn ),
α (xn − L ) = stat2 o(an ) (α ∈ R ) and
(xn − L1 )(yn − L2 ) = stat2 o(an bn ), where
cn = max {an , bn }. n∈N
N.L. Braha et al. / Applied Mathematics and Computation 266 (2015) 675–686
683
Proof. We only prove just the first assertion of Lemma 1; the other assertions can be proved similarly. For a given > 0, let us set
A1 = {k : k λn − λn−1
and
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) + (λk yk − 2λk−1 yk−1 + λk−2 yk−2 ) − (L1 + L2 )| },
and
|(λk xk − 2λk−1 xk−1 + λk−2 xk−2 ) − L1 |
and
|(λk yk − 2λk−1 yk−1 + λk−2 yk−2 ) − L2 |
A2 = k : k λn − λn−1 and
A3 = k : k λn − λn−1
2
2
.
Then we observe that A1 ⊂ A2 ∪ A3 . Moreover, since
cn = max {an , bn }, n∈N
we obtain
|A1 | |A2 | |A2 | + . (λn − λn−1 ) · cn (λn − λn−1 ) · cn (λn − λn−1 ) · cn
(21)
Now, by taking the limit as n → ∞ in (21) and using the hypothesis of Lemma 1, we conclude that
lim
n→∞
|A1 | = 0, (λn − λn−1 ) · cn
which proves the first assertion of Lemma 1. We now recall that the modulus of continuity for a function f (x ) ∈ C2π (R ) is defined as follows:
ω ( f, δ ) = sup | f (x + h ) − f (x )|. |h|<δ
We consider the following two cases: (A) If |x − y| < δ, then
| f (x ) − f (y )| ω ( f, δ ) and (B) If |x − y| > δ, then
| f (x ) − f (y )| ω ( f, δ ) ω ( f, δ ) ·
|x − y| . δ
From these last two relations, we find for any value of |x − y| that
| f (x ) − f (y )| ω ( f, δ )
|x − y| 2
+1 .
We state and prove the following result. Theorem 5. Let (Bn ) be a sequence of positive linear operators from C[a, b] into C[a, b]. Suppose that
||Bn (1, x ) − 1||∞ = stat2 o(an ) and
ω ( f, λn ) = stat2 o(bn ), where
λn =
Bn ( ψ , x )
and
ψ ≡ ψ (t, x ) = (e−t − e−x )2 .
Then, for all f ∈ C[a, b] and x ∈ [a, b], we have
||Bn ( f, x ) − f (x )||∞ = stat2 − o(cn ), where
cn = max {an , bn }. n∈N
(22)
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N.L. Braha et al. / Applied Mathematics and Computation 266 (2015) 675–686
Proof. Let f ∈ C[a, b] and x ∈ [a, b]. From the relations (13) and (22), we get the following estimation:
|Bn ( f, x ) − f (x )| |Bn (| f (y ) − f (x )|, x )| + | f (x )| · |Bn (1, x ) − 1| |x − y| Bn + 1, x ω ( f, δ ) + | f (x )| · |Bn (1, x ) − 1| δ
M Bn 1 + (e−t − e−x )2 , x ω ( f, δ ) + | f (x )| · |Bn (1, x ) − 1| δ
M Bn (1, x ) + Bn (ψ , x ) ω ( f, δ ) + | f (x )| · |Bn (1, x ) − 1|. δ If we put
δ=
λn 2 M
=
Bn ( ψ , x ) M
in the last relation, we obtain
||Bn ( f, x ) − f (x )||∞ || f ||∞ ||Bn (1, x ) − 1||∞ + ω ( f, λn ) + ω ( f, λn )||Bn (1, x ) − 1||∞ C {||Bn (1, x ) − 1||∞ + ω ( f, λn ) + ω ( f, λn )||Bn (1, x ) − 1||∞ }, where
C = max
|| f ||∞ , 1 .
Now, upon replacing Bk (., x) by
2 =
n 1 (λk Bk (., x ) − 2λk−1 Bk−1 (., x ) + λk−2 Bk−2 (., x )), λn − λn−1 k=0
we get
||2 ( f, x ) − f (x )||∞ C ||2 (1, x ) − 1||∞ + ω ( f, λn ) + ω ( f, λn )||2 (1, x ) − 1||∞ .
The proof of Theorem 5 now follows readily from the hypotheses and Lemma 1. 5. Voronovskaya type theorems In this section, we show that the positive linear operators Ln defined in Example 2 satisfy a Voronovskaja type property in the
2 (stat) sense. We first prove the following lemma: Lemma 2. For x ∈ [0, 1] and (y ) = y − x,
n2 Ln ( 4 ) ∼ 3x2 (1 + x )2 (2 (stat )) on
[0, 1].
Proof. After some calculations, we get
x(1 + x )(6x2 + 6x + 1 ) 3x2 (1 + x )2 n Ln ( ) = n (1 + xn ) + , n2 n3 2
4
2
which shows that
2 n Ln ( 4 ) − 3x2 (1 + x )2 2 x ( 1 + x )( 6x + 6x + 1 ) 2 2 2 2 − 3x (1 + x ) = (1 + xn )3x (1 + x ) + (1 + xn ) n
2 x(1 + x )(6x2 + 6x + 1 ) 3x2 (1 + x )2 |(1 + xn ) − 1| + (1 + xn ) → 0 (stat ) on [0, 1]. n
This evidently completes proof of Lemma 2. The following Voronovskaya type theorem for operators Ln given in Example 2 holds true. Theorem 6. For every f ∈ C[0, 1] such that f , f ∈ C[0, 1], then
n Ln ( f ) − f ∼ on [0, 1].
1 (x + x2 ) f (x ) 2
2 (stat )
(23)
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685
Proof. Let us suppose that f , f ∈ C[0, 1] and x ∈ [0, 1]. Define
f (y ) − f (x ) − (y − x ) f (x ) − 12 (y − x )2 f (x ) ( y − x )2 0
ψx (y ) =
(x = y ) ( x = y ).
Then ψx (x ) = 0 and ψ x ∈ C[0, 1]. By Taylor’s formula, we get
f (y ) = f (x ) + (y − x ) f (x ) +
1 (y − x ) f (x ) + (y − x )2 ψx (y ). 2
(24)
Knowing that
Ln (1, x ) = (1 + xn ); Ln (y − x ), x = 0 and
Ln (y − x )2 , x = (1 + xn )
x (1 + x ) , n
and after operating on both sides of the relation (24) by the operator Ln , we obtain
f (x ) x(1 + x ) (1 + xn ) + (1 + xn )Vn ( 2 ψx , x ), 2 n
Ln ( f ) = f ( x ) + xn f ( x ) + which yields
1 n[Ln ( f ) − f (x )] − (x + x2 ) f (x ) nxn | f (x )| + xn | f (x )| + n(1 + xn )Vn ( 2 ψx , x ) 2
and
1 n[Ln ( f ) − f (x )] − 2 (x + x2 ) f (x ) nxn M + n(1 + xn )Vn ( 2 ψx , x ),
(25)
where
(y ) = y − x and M = || f ||C[0,1] + || f ||C[0,1] . After application of the Cuachy–Schwarz inequality in the second term of the right-hand side of the relation (25), we find that
nVn ( 2 ψx , x ) [n2Vn ( 4 , x )] 2 · [Vn (ψx , x )] 2 . 1
1
(26)
2 Also, by putting ηx (y ) = ψx (y ) , we see that ηx (x ) = 0 and ηx ( · ) ∈ C[0, 1]. Clearly, from Remark 2, it follows that
Ln (ηx ) → 0 2 (stat )
on
[0, 1].
(27)
Now, from the relations (26) and (27) and Lemma 2, we have
Ln ( 2 ψx , x ) → 0 2 (stat )
on [0, 1].
(28)
For a given > 0, we define the following sets:
An (x, ) = k : k λn − λn−1
A1,n (x, ) = k : k λn − λn−1 and
1 k(Vk ( f, x ) − f (x )) − 2 (x + x2 ) f (x ) , and |kxn |
and
A2,n (x, ) = k : k λn − λn−1
2M
and
|kVn ( 2 ψx , x )|
. 2
From these last relations, we obtain
An (x, ) A1,n (x, ) + A2,n (x, ) and
2 (An ·, ) 2 (A1,n ·, ) + 2 A2,n (·, ) .
(29)
Moreover, by the definition of the sequence X = (xn ), we get
nxn → 0
2 (stat )
on
[0, 1].
(30)
Now, from the relations (28) and (30), the right-hand side of the relation (29) tends to zero as n → ∞. Therefore, we have
stat limn→∞ 2 An (·, ) = 0, which proves that
n Ln ( f ) − f ∼
1 (x + x2 ) f (x ) 2 (stat ) on 2
[0, 1].
We have thus completed our proof of Theorem 6. Remark 3. Since the sequence X = (xn ) given in Example 1 is not statistically convergent, we conclude that the operators (Ln ) defined in Example 2 do not satisfy a Voronovskaya type theorem in the usual sense.
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References [1] F. Altomare, Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory 5 (2010) 92–164. [2] B.D. Boyanov, V.M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 14 (62) (1970) 9–13. [3] N.L. Braha, T. Mansour, On λ2 -strong convergence of numerical sequences and fourier series, Acta Math. Hungar. 141 (2013) 113–126. [4] N.L. Braha, H.M. Srivastava, S.A. Mohiuddine, A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la vallée poussin mean, Appl. Math. Comput. 228 (2014) 162–169. [5] O.H.H. Edely, S.A. Mohiuddine, A.K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett. 23 (2010) 1382–1387. [6] E. Erkus, ¸ O. Duman, H.M. Srivastava, Statistical approximation of certain positive linear operators constructed by means of the Chan–Chyan–Srivastava polynomials, Appl. Math. Comput. 182 (2006) 213–222. [7] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244. [8] P.P. Korovkin, Convergence of linear positive operators in the spaces of continuous functions, Dokl. Akad. Nauk. SSSR (N.S.) 90 (1953) 961–964. (in Russian) [9] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corporation, Delhi, 1960. [10] S.A. Mohiuddine, A. Alotaibi, M. Mursaleen, Statistical summability (c, 1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012) 1–8. Article ID 2012:172 [11] M. Mursaleen, A. Alotaibi, Statistical lacunary summability and a Korovkin type approximation theorem, Ann. Univ. Ferrara 57 (2011) 373–381. [12] M. Mursaleen, V. Karakaya, M. Erturk, F. Gursoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput. 218 (2012) 9132–9137. [13] M. Mursaleen, A. Khan, H.M. Srivastava, K.S. Nisar, Operators constructed by means of q-Lagrange polynomials and a-statistical approximation, Appl. Math. Comput. 219 (2013) 6911–6918. [14] M.A. Özarslan, O. Duman, H.M. Srivastava, Statistical approximation results for Kantorovich-type operators involving some special polynomials, Math. Comput. Modelling 48 (2008) 388–401. [15] R. P˘alt˘anea, Approximation Theory Using Positive Linear Operators, Birkhäuser, Boston, Basel and Berlin, 2004. [16] H.M. Srivastava, M. Mursaleen, A. Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Modelling 55 (2012) 2040–2051. [17] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951) 73–74. [18] J.J. Wang, Z.B. Xu, Approximation with Jacobi weights by Baskakov operators, Taiwanese J. Math. 13 (2009) 157–168.